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A note on internality of certain differential systems

Partha Kumbhakar, Varadharaj Ravi Srinivasan

Abstract

We prove two results, generalizing certain theorems by Jin and Moosa, on the internality of the system of differential equations \begin{equation*} \begin{aligned} x' &= f(x)\\ y' &= g(x)y,\\ \end{aligned} \end{equation*}where $f$ and $g$ are rational functions in one variable.

A note on internality of certain differential systems

Abstract

We prove two results, generalizing certain theorems by Jin and Moosa, on the internality of the system of differential equations \begin{equation*} \begin{aligned} x' &= f(x)\\ y' &= g(x)y,\\ \end{aligned} \end{equation*}where and are rational functions in one variable.

Paper Structure

This paper contains 4 theorems, 14 equations.

Key Result

Proposition 1

Let be a system almost internal to the constants. Then, there exists a finitely generated differential field extension $M$ of $K$ such that for any generic solution $(x,y)$ over $M$ of the systemGeneric solutions over $M$ of the system always exist. To see this, consider the rational function field $M(s

Theorems & Definitions (9)

  • Proposition 1
  • proof
  • Theorem 2
  • proof
  • Remark 3
  • Theorem 4
  • proof
  • Theorem 5
  • proof